Given: X and Y, independent, continuous

**uniform** random variables on the interval (0,1)

Goal: Find the density function of$\displaystyle \frac{X}{X+Y}$

My approach:

Use the transformation technique (as in point 5 of this wikipedia page

Integration by substitution - Wikipedia, the free encyclopedia).

Define $\displaystyle $Z_{1} = \frac{X}{X+Y}$$. Define $\displaystyle $Z_{2} = X$$.

I find that the joint density of$\displaystyle $Z_{1}$$ and $\displaystyle $Z_{2}$$ is

$\displaystyle $f_{Z_{1},Z_{2}}(z_{1},z_{2}) = \frac{Z_{2}}{{Z_{1}}^{2}}$$

And this is where i get stuck... I tried integrating $\displaystyle Z_{2}$ out of it but nothing reasonable comes out (not a density function anyway). So two questions:

- Did I find the joint density correct?
- I tried integrating$\displaystyle Z_{2}$ over the interval$\displaystyle Z_{1}$ to 1, is this correct?
- What should be next step?

Thanks for any help!